Mappings of Adjoint Linear Operators

adjoint-operatorslinear algebramatricesmatrix equationsnumerical linear algebra

Let
$$
\begin{align*}
\mathsf{A}:~ \mathbb{R}^n
& \rightarrow \mathbb{R}^m \\
\boldsymbol{x}
& \mapsto \boldsymbol{y} = \boldsymbol{A} \boldsymbol{x}.
\end{align*}
$$

The adjoint of the linear operator $\mathsf{A}$, denoted by $\mathsf{A}^*$, is defined via
$$
\begin{align*}
\left\langle \mathsf{A}(\boldsymbol{x}),\boldsymbol{y} \right\rangle
& = \left\langle \boldsymbol{x},\mathsf{A}^*(\boldsymbol{y}) \right\rangle
\end{align*}
$$
for all $\boldsymbol{x} \in \mathbb{R}^n$ and $\boldsymbol{y} \in \mathbb{R}^m$.

$\mathsf{A}^{*}(\boldsymbol{y}) = \boldsymbol{A}^{\mathsf{T}} \boldsymbol{y}$
Consider the linear map given by
$$
\begin{align*}
& \boldsymbol{Y} = \mathsf{C}(\boldsymbol{X}) = \boldsymbol{A} \boldsymbol{X} \boldsymbol{B}
\end{align*}
$$

  1. What would the adjoint of $\mathsf{C}$ be interms of matrices $\boldsymbol{A}$ and $\boldsymbol{B}$.

  2. Let $\boldsymbol{Z} = \mathsf{C}^*( \mathsf{C} (\boldsymbol{X}) )$. How can I compute the closed form of Z interms A,B?

I have tried approaching this problem using the given definition for a vector and not a matrix but I am going in circles.

What would be a good start here?

Best Answer

Take the matrices $A,B$ from the definition of $\mathsf C$ to be of sizes $p \times m, n \times q$ respectively. With that, $\mathsf C$ is a linear map $\mathsf C:\Bbb R^{m \times n} \to \Bbb R^{p \times q}$.

As you've obliquely noted, $\mathsf C$ takes matrices and produces matrices, whereas the map $\mathsf A$ from your definition of the adjoint takes vectors and produces vectors. If we are to apply your definition to $\mathsf C$, we'll need to extend your definition to this new context. To that end, we'll define the following inner product (known as the "Frobenius" or "Hilbert-Schmidt" inner product) over $\Bbb R^{m \times n}$ and $\Bbb R^{p \times q}$. For matrices $X,Y$ of the same shape, define $$ \langle X,Y \rangle = \operatorname{tr}(X^TY) = \sum_{i,j} x_{ij} y_{ij}. $$ To compute the adjoint of $\mathsf C$, note that for $m \times n$ matrices $X,Y$, we have \begin{align} \langle \mathsf C(X),Y \rangle &= \langle AXB,Y \rangle = \operatorname{tr}((AXB)^TY) = \operatorname{tr}(B^TX^TA^TY) \\ & = \operatorname{tr}(X^TA^TYB^T) = \langle X, A^TYB^T \rangle. \end{align} It follows that the adjoint of $\mathsf C$ is given by $\mathsf C^*(Y) = A^TYB^T$. From there, we can compute $$ \mathsf C^*(\mathsf C(X)) = A^TAXB^TB. $$

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